David J. Green

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Cohomology of group number 143 of order 128

About the group

Ring generators

Ring relations

Completion information

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General information on the group

• The group has 2 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t5  +  t2  +  1

  (t  −  1)2 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

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Ring generators

The cohomology ring has 12 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_2_1, a nilpotent element of degree 2
4. a_2_2, a nilpotent element of degree 2
5. a_3_2, a nilpotent element of degree 3
6. a_3_3, a nilpotent element of degree 3
7. b_4_3, an element of degree 4
8. a_5_2, a nilpotent element of degree 5
9. a_5_4, a nilpotent element of degree 5
10. b_6_5, an element of degree 6
11. a_7_5, a nilpotent element of degree 7
12. c_8_6, a Duflot regular element of degree 8

About the group

Ring generators

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Ring relations

There are 47 minimal relations of maximal degree 14:

1. a_1_0²
2. a_1_0·a_1_1
3. a_2_1·a_1_0
4. a_2_2·a_1_1 + a_2_1·a_1_1
5. a_2_2·a_1_0 + a_2_1·a_1_1
6. a_1_1⁴
7. a_2_2² + a_2_1·a_2_2
8. a_1_0·a_3_2
9. a_1_1·a_3_3 + a_2_2²
10. a_1_0·a_3_3 + a_2_1²
11. a_2_2·a_3_2
12. a_2_1·a_3_2
13. a_2_1·a_3_3 + a_1_1²·a_3_2
14. b_4_3·a_1_0
15. a_3_2·a_3_3 + a_1_1³·a_3_2
16. a_2_1·b_4_3 + a_3_3²
17. a_3_2² + b_4_3·a_1_1²
18. a_1_1·a_5_2 + a_1_1³·a_3_2
19. a_1_0·a_5_2
20. a_1_0·a_5_4 + a_1_1³·a_3_2
21. a_2_1·a_5_2 + b_4_3·a_1_1³
22. a_2_2·a_5_4
23. a_2_2·a_5_2 + a_2_1·a_5_4 + b_4_3·a_1_1³
24. b_6_5·a_1_1 + b_4_3·a_3_2 + a_2_2·a_5_2 + a_1_1²·a_5_4
25. b_6_5·a_1_0
26. a_3_2·a_5_2 + a_2_2·a_3_3²
27. a_2_2·a_3_3² + a_1_1³·a_5_4
28. a_2_2·b_6_5 + a_3_3·a_5_4 + a_3_3·a_5_2 + a_2_2·a_3_3²
29. a_2_1·b_6_5 + a_3_3·a_5_2 + a_2_2·a_3_3²
30. a_3_2·a_5_4 + a_1_1·a_7_5 + b_4_3·a_1_1·a_3_2 + a_2_2·a_3_3²
31. a_1_0·a_7_5
32. b_6_5·a_3_3 + b_6_5·a_3_2 + b_4_3·a_5_2 + b_4_3²·a_1_1 + b_4_3·a_1_1²·a_3_2
33. a_2_2·a_7_5
34. a_2_2·b_4_3·a_3_3 + a_2_1·a_7_5 + b_4_3·a_1_1²·a_3_2
35. b_6_5·a_3_3 + b_4_3·a_5_2 + a_2_2·b_4_3·a_3_3 + a_1_1²·a_7_5 + b_4_3·a_1_1²·a_3_2
36. a_5_2² + b_4_3·a_3_3²
37. a_2_2·b_4_3² + a_5_2·a_5_4 + b_4_3·a_3_3²
38. a_2_2·b_4_3² + a_3_3·a_7_5 + b_4_3·a_3_3² + a_1_1³·a_7_5
39. a_3_2·a_7_5 + b_4_3·a_1_1·a_5_4 + b_4_3²·a_1_1²
40. a_2_2·b_4_3² + a_5_4² + b_4_3·a_3_3² + b_4_3·a_1_1·a_5_4 + c_8_6·a_1_1²
41. b_6_5·a_5_2 + b_4_3²·a_3_3 + a_3_3²·a_5_4 + b_4_3·a_1_1²·a_5_4
42. b_6_5·a_5_4 + b_4_3·a_7_5 + b_4_3²·a_3_2 + a_2_1·c_8_6·a_1_1 + c_8_6·a_1_1³
43. a_5_2·a_7_5 + b_4_3·a_3_3·a_5_4 + b_4_3·a_1_1³·a_5_4
44. a_5_4·a_7_5 + b_4_3·a_3_3·a_5_4 + b_4_3·a_1_1³·a_5_4 + c_8_6·a_1_1·a_3_2
45. b_6_5² + b_4_3³ + b_4_3·a_3_3·a_5_2 + b_4_3²·a_1_1·a_3_2 + a_2_1²·c_8_6
46. b_6_5·a_7_5 + b_4_3²·a_5_4 + b_4_3³·a_1_1 + c_8_6·a_1_1²·a_3_2
47. a_7_5² + b_4_3·a_3_3·a_7_5 + b_4_3²·a_1_1·a_5_4 + b_4_3³·a_1_1²
       + b_4_3·a_1_1³·a_7_5 + b_4_3·c_8_6·a_1_1²

About the group

Ring generators

Ring relations

Completion information

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Back to groups of order 128

Data used for Benson′s test

• Benson′s completion test succeeded in degree 14.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_8_6, a Duflot regular element of degree 8
  2. b_4_3, an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, 6, 10].
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. a_2_2 → 0, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. b_4_3 → 0, an element of degree 4
8. a_5_2 → 0, an element of degree 5
9. a_5_4 → 0, an element of degree 5
10. b_6_5 → 0, an element of degree 6
11. a_7_5 → 0, an element of degree 7
12. c_8_6 → c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. a_2_2 → 0, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. b_4_3 → c_1_1⁴, an element of degree 4
8. a_5_2 → 0, an element of degree 5
9. a_5_4 → 0, an element of degree 5
10. b_6_5 → c_1_1⁶, an element of degree 6
11. a_7_5 → 0, an element of degree 7
12. c_8_6 → c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128